| L(s) = 1 | + (−0.115 − 2.23i)5-s + 0.448i·7-s − 1.79·11-s + 3.55i·13-s + 1.08i·17-s − 0.502·19-s − 1.38i·23-s + (−4.97 + 0.514i)25-s − 3.33·29-s + 2.17·31-s + (1.00 − 0.0517i)35-s − i·37-s + 0.833·41-s − 11.0i·43-s + 5.06i·47-s + ⋯ |
| L(s) = 1 | + (−0.0515 − 0.998i)5-s + 0.169i·7-s − 0.541·11-s + 0.985i·13-s + 0.262i·17-s − 0.115·19-s − 0.288i·23-s + (−0.994 + 0.102i)25-s − 0.618·29-s + 0.390·31-s + (0.169 − 0.00874i)35-s − 0.164i·37-s + 0.130·41-s − 1.68i·43-s + 0.739i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.609753038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609753038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.115 + 2.23i)T \) |
| 37 | \( 1 + iT \) |
| good | 7 | \( 1 - 0.448iT - 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 13 | \( 1 - 3.55iT - 13T^{2} \) |
| 17 | \( 1 - 1.08iT - 17T^{2} \) |
| 19 | \( 1 + 0.502T + 19T^{2} \) |
| 23 | \( 1 + 1.38iT - 23T^{2} \) |
| 29 | \( 1 + 3.33T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 41 | \( 1 - 0.833T + 41T^{2} \) |
| 43 | \( 1 + 11.0iT - 43T^{2} \) |
| 47 | \( 1 - 5.06iT - 47T^{2} \) |
| 53 | \( 1 + 2.12iT - 53T^{2} \) |
| 59 | \( 1 - 8.02T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.15iT - 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 - 14.3iT - 73T^{2} \) |
| 79 | \( 1 + 8.21T + 79T^{2} \) |
| 83 | \( 1 - 13.0iT - 83T^{2} \) |
| 89 | \( 1 + 4.73T + 89T^{2} \) |
| 97 | \( 1 - 15.4iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.298795765437125910537357782902, −7.23917334357175188243166053625, −6.70085692390640060196526323484, −5.58919509167780596095329686359, −5.34203088615764693762295405546, −4.25251061769887292935495059041, −3.91245498024311458618248111719, −2.57655942290311837755751366294, −1.84142547492890172228316418547, −0.72702023487486980926544363571,
0.56712480625478594934406768245, 1.97333993978143239638921332188, 2.84869584275778506160297849179, 3.41046107367274049766522565886, 4.31962899961291194587819291120, 5.25884530131480098738343149943, 5.88459736908105783354212198095, 6.62411123041203568339519458708, 7.37299302221521403420592679209, 7.82919234453713639840778461387
